How do I make a sphere with 5 flat sides?
Another method:


Add a sphere (both types will work).
Add a cylinder with five verts:


Add a boolean modifier to the sphere and select the cylinder as the object:



Scaling the cylinder on the X and Y (SShiftZ) axes will change the radii of the flatten areas.
Apply the modifier and select the flat faces (the fastest was is probably by selecting one and pressing ShiftG> Area):


Press AltP and the select the center vertices:


Scale the vertices out (S). Then optionally set shading to smooth and add an edge split modifier: